Question

Coefficient of determination measures 

(A) Correlation between the dependent and independent variables. 
(B) The residual sum of squares as a proportion of the total sum of squares. 
(C) The explained sum of squares as a proportion of the total sum of squares. 
(D) How well the sample regression fits the data. 
Choose the correct answer from the options given below:
 

• A. (A), (B) and (D) only
• B. (A), (C) and (D) only
• C. (A), (B), (C) and (D)
• D. (C) and (D) only

Hint:

The coefficient of determination (\( R^2 \)) measures the proportion of variance explained by the regression model. Higher \( R^2 \) values indicate better model fit.

Answer 1

The Correct Option is B

Step 1: Define the coefficient of determination.
The coefficient of determination, symbolized as \( R^2 \), quantifies the portion of the dependent variable's variance attributable to the independent variables. It is a critical indicator of a regression model's explanatory power for the data.

Step 2: Evaluate the provided options.
- (A) Correlation between the dependent and independent variables: This is partially accurate. While \( R^2 \) is derived from the square of the correlation coefficient, it is not identical to the correlation itself.
- (B) The residual sum of squares as a proportion of the total sum of squares: This is inaccurate. \( R^2 \) reflects the proportion of explained variation, not residual variation, relative to the total variation.
- (C) The explained sum of squares as a proportion of the total sum of squares: This is accurate. The coefficient of determination is computed as the ratio of the explained variation to the total variation in the dependent variable.
- (D) How well the sample regression fits the data: This is accurate. \( R^2 \) serves as a metric for the goodness-of-fit of the regression model to the observed data.

Step 3: State the conclusion.
Options (A), (C), and (D) correctly describe the coefficient of determination.